There is a large number of issues there that need to be addressed. I feel like there was already a thread on this exact situation before though.
First tying. A horse should know how to stand before you encourage them to move faster than a walk. If they can't stop, don't ask for speed. To teach her to stand, I teach ground tying before hard tying. I put two leads on the horse, one on the ground (a short one they can't trip over if something awful happened. Then tell the horse "stand" take a step back, if the horse moves even 1 inch in any direction, immediately place her feet back where they belong.
This leads me to the next issue. She's a serious space invader - in order to teach her to stand and to respect your space she needs to be taught to yield. She needs to learn to move her hing and front end away with gentle pressure. Apply a small amount of pressure on her hip or shoulder, wait and the moment she moves that part of her away release the pressure and rub the spot out. Do this until she will turn either part of her away with just the forward motion of your hand. Then use the same skill on the top of her nose (where the top of a flat halter would lie) and ask for back up. Stand in front of her, forceably step into her space and apply gentle pressure to the noseband of her halter wait until she backs up, release and praise (praise can be as simple as a 'good' or a rub or just nothing, whatever this horse wants). Once she is backing and yielding her front and hind end easily teach her to put her head down with poll pressure, apply gentle pressure on the lead rope so that there's only pull on the horse's poll, wait, relieve pressure and praise when she moves her head down even a little. If she is non-responsive to this put the rope up over her poll where the earband of the halter is, gentle pressure and wait.
Teaching a horse to put their head down is a great way to help flighty horses shift out of flight mode.
Until she stands when there's a rope on the ground and you can walk all the way around her without her moving an inch and can lead at the walk without invading your space or being rude do not try to trot her in hand. These skills are not individual they are all built on the last. You can't have a solid trot without a solid walk. You need to start this horse from step 1 - no skipping ahead.
Good luck let me know if you need something further explained. I'd also suggest finding a trainer who could perhaps come and show you in person. They can better point out the finesse of training that the internet just can't carry.
ETA: As for the head shaking it could be a number of things. Most obviously is bugs. There are bugs too small for people to see that buzz horse's faces and drive them crazy. Rub some bug spray on her forhead and outside of her ears careful not to get any in her eyes. If she has bug spray on and is still doing it, it could be an ear infection or ear mites or any number of other issues you should have the vet check out.
ETA x2: Darrin brought up a good point- the forlock! I forgot about that, my pony needs his for-lock braided and tucked under his halter because he can't see through it - is her's long or thick? When my pony runs out in his field he throws his head up to get it out of his eyes before he runs off.
Last edited by PunksTank; 09-08-2012 at 01:06 PM.